Coarse pseudo-differential calculus and index theory on manifolds with a tangent Lie structure

TL;DR

This paper develops a simplified pseudo-differential calculus for manifolds with tangent Lie structures, enabling an index theorem where indices are elements of K-homology groups, extending classical analysis to more general geometric contexts.

Contribution

It introduces a coarse pseudo-differential calculus on manifolds with tangent Lie structures and proves an associated index theorem with K-homology group values.

Findings

Established a coarse PDO calculus for manifolds with tangent Lie structures.

Proved an index theorem for h-elliptic operators with indices in K-homology.

Extended classical pseudo-differential analysis to filtered and tangent Lie structured manifolds.

Abstract

We introduce a simplified (coarse) version of pseudo-differential calculus for operators of order zero on complete Riemannian manifolds. This calculus works for the usual Hormander (1,0) class of operators, as well as for pseudo-differential operators on filtered manifolds. In fact, we develop the coarse PDO calculus on a more general class of manifolds which we call manifolds with a tangent Lie structure. We prove an index theorem for `h-elliptic' operators where the index is not just an integer, but an element of the K-homology group of the manifold.